Simulation of a diffusion model describingdynamic price information

Benjamin Yiwen Färber

May 7, 2014

Contents

1 Statistical stock market 31.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.2 Specification of the model . . . . . . . . . . . . . . . . . . . . . . 31.3 Relevance to the financial market . . . . . . . . . . . . . . . . . . 41.4 Agent’s behaviour . . . . . . . . . . . . . . . . . . . . . . . . . . 4

List of Figures 5

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Chapter 1

Statistical stock market

1.1 IntroductionStatistical physics describes phenomena consisting of very large numbers of par-ticles or systems. One can think here of a classical or a quantum gas consistingof very large numbers of molecules or atoms. To accurately describe this sys-tem, it is isolated from the environment as interactions with the environmentleads to complex behaviour. Statistical physics does not need to be restrictedto the description of a gas but may be applied to dynamic phenomena like theformation of gas bubbles in a boiling liquid. Lately it has been applied to othernon-physical contexts. The field of econophysics has emerged from address-ing statistical physics methods and ideas to economic questions. The model ofBrownian motion has been used to study prices in stock markets. Stock mar-ket bubbles and crashes were investigated in the framework of the Ising modelof magnetism. The main aspect in Econophysics are complex systems. Theirbehaviour can be described as a result of a collective effect involving many in-teracting particles. Forming an analogy of the stock market with molecularsystems, interactions of individual agents with their surroundings can be com-pared to an ensemble of decisions in a statistical steady state, described by pricequotations in a stock market.

1.2 Specification of the modelA detailed definition of the model will be discussed here. The spatial distribu-tion of particles is modeled on a quadratic lattice G = {0, 1, . . . , L− 1} of sizeL × L with periodic boundary conditions. Two types of particles are denotedas sellers and buyers. The sellers are fixed at their positions on the lattice andare characterised, apart from their coordinates, by a certain price value p, atwhich they sell. The buyers on the other hand diffuse over the lattice. They arecharacterised by a value θ, which is their valuation. The difference between pand θ defines the probability that a sale takes place. In the lattice gas model theprices p and valuations θ are constant. Later, turning to a multi-agent system,sellers change their prices p through a learning algorithm. The sellers might re-spond to the aggregated individual demand similarly to the price settings foundin Brownian motion models for the stock market. This might be an equilibrium

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Figure 1.1: Sellers act as agents on a lattice, on which buyers, acting as agents,move randomly, and occasionally buy at a seller when they hit the seller on alattice vertice.

process. As buyers are subject to the local neighbourhoods of individual sellersthe solution of the value function θ which is the expected equilibrium utility ofstaying in the neighbourhood may be in direct analogy to the distribution ofthe price variation δp.

1.3 Relevance to the financial marketThe information affecting the dynamics of the price of a financial asset plays asignificant role in real financial markets. It may be of concern in this model assellers act upon information regarding their profit.

1.4 Agent’s behaviourLearning is goal directed and can be modeled as a trial and error search. Theagent’s interaction with the environment is based on the valuation of the stateof the environment called the value function V (s) with s the state of the en-vironment. The agent has the ability to plan the next action according to itsmodel of the environment. The value function may be implemented in form ofa temporal difference learning function,

V (s)← V (s) + α (V (s′)− V (s)) (1.1)

where α is the step-size parameter. The value of a state s is updated to avalue regarding the difference of the values of the state s′ and s.

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List of Figures

1.1 Sellers act as agents on a lattice, on which buyers, acting asagents, move randomly, and occasionally buy at a seller whenthey hit the seller on a lattice vertice. . . . . . . . . . . . . . . . 4

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Bibliography

[Sta04] Mantegna, R. N., Stanley, E., "An Introduction to Econophysics. Cor-releations and Complexity in Finance."Cambridge University Press 2004

[Osb59] Osborne, M.F.M., "Browninan motion in the stock market" OperationsResearch, Vol. 7 (1959), pp. 215-219

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